A342154 Number of partitions of n^5 into two positive squares.

0, 0, 1, 0, 0, 3, 0, 0, 1, 0, 3, 0, 0, 3, 0, 0, 0, 3, 1, 0, 3, 0, 0, 0, 0, 5, 3, 0, 0, 3, 0, 0, 1, 0, 3, 0, 0, 3, 0, 0, 3, 3, 0, 0, 0, 3, 0, 0, 0, 0, 6, 0, 3, 3, 0, 0, 0, 0, 3, 0, 0, 3, 0, 0, 0, 18, 0, 0, 3, 0, 0, 0, 1, 3, 3, 0, 0, 0, 0, 0, 3, 0, 3, 0, 0, 18, 0, 0, 0, 3, 3, 0, 0, 0, 0, 0, 0, 3, 1, 0, 5, 3, 0, 0, 3, 0

Offset: 0

Author

Seiichi Manyama, Mar 02 2021

Keywords

• nonn
• easy

Comments

a(n) > 0 if and only if n is in A000404. - Robert Israel, Mar 03 2021

Examples

			2^5 = 32 = 4^2 + 4^2. So a(2) = 1.
5^5 = 3125 = 10^2 + 55^2 = 25^2 + 50^2 = 38^2 + 41^2. So a(5) = 3.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..3050
• Index entries for sequences related to sums of squares

Crossrefs

Cf. A000404, A000584, A025426, A046080, A084888.

Programs

Maple

f:= proc(n) local x,y,S;
      S:= map(t -> subs(t,[x,y]),[isolve(x^2+y^2=n^5)]);
      nops(select(t -> t[1] >= t[2] and t[2] > 0, S))
end proc:
map(f, [$0..200]); # Robert Israel, Mar 03 2021

PARI

a(n) = my(cnt=0, m=n^5); for(k=1, sqrt(m/2), l=m-k*k; if(l>0&&issquare(l), cnt++)); cnt;

Formula

a(n) = A025426(A000584(n)).